The epispirals are the projections of the geodesics of the cone of revolution on a plane perpendicular to its axis, the half-angle of the cone being equal to .

The Poinsot spirals are solutions of the problem that consists in determining the trajectories in space of a massive point subject to a force centred on O proportional to (this force is, thanks to the Binet formula, proportional to
which is here ,
with ); the other solutions are the Poinsot spirals, with intermediary case the hyperbolic spiral, see this link.

Is it because the base pattern reminds of the awn of an ear of wheat (épi in French) that the curve has this name?